A Game-theoretic Approach Towards Collaborative Coded Computation Offloading
Jer Shyuan Ng, Wei Yang Bryan Lim, Zehui Xiong, Dusit Niyato, Cyril Leung, Dong In Kim, Junshan Zhang, Qiang Yang
11 A Game-theoretic Approach TowardsCollaborative Coded Computation Offloading
Jer Shyuan Ng, Wei Yang Bryan Lim, Zehui Xiong, Dusit Niyato,
Fellow, IEEE , Cyril Leung,Dong In Kim,
Fellow, IEEE , Junshan Zhang,
Fellow, IEEE , Qiang Yang,
Fellow, IEEE
Abstract —Coded distributed computing (CDC) has emerged as a promising approach because it enables computation tasks to becarried out in a distributed manner while mitigating straggler effects, which often account for the long overall completion times.Specifically, by using polynomial codes, computed results from only a subset of edge servers can be used to reconstruct the finalresult. However, incentive issues have not been studied systematically for the edge servers to complete the CDC tasks. In this paper,we propose a tractable two-level game-theoretic approach to incentivize the edge servers to complete the CDC tasks. Specifically, inthe lower level, a hedonic coalition formation game is formulated where the edge servers share their resources within their coalitions.By forming coalitions, the edge servers have more Central Processing Unit (CPU) power to complete the computation tasks. In theupper level, given the CPU power of the coalitions of edge servers, an all-pay auction is designed to incentivize the edge servers toparticipate in the CDC tasks. In the all-pay auction, the bids of the edge servers are represented by the allocation of their CPU power tothe CDC tasks. The all-pay auction is designed to maximize the utility of the cloud server by determining the allocation of rewards tothe winners. Simulation results show that the edge servers are incentivized to allocate more CPU power when multiple rewards areoffered, i.e., there are multiple winners, instead of rewarding only the edge server with the largest CPU power allocation. Besides, theutility of the cloud server is maximized when it offers multiple homogeneous rewards, instead of heterogeneous rewards.
Index Terms —Coded distributed computing, straggler effects mitigation, hedonic game, all-pay auction, Bayesian Nash equilibrium (cid:70)
NTRODUCTION C OUPLED with reliable wireless communication tech-nologies, IoT devices can serve as important sources ofsensor data for Artificial Intelligence (AI) technologies to beleveraged, towards the development of data-driven applica-tions [1]. In particular, many machine learning models aredeveloped to monitor various large-scale physical phenom-ena for smart city applications, such as prediction of roadconditions [2], air quality monitoring [3] and tracking ofmedical conditions [4]. Edge computing [5] has emerged as apromising approach that extends cloud computing servicesto the edge of the networks. In particular, by leveragingon the computational capabilities, e.g., Central ProcessingUnit (CPU) power, of the edge servers, e.g., base stationsand edge devices, e.g., laptops and tablets, the cloud servercan offload its computation tasks to the edge servers anddevices.However, there are several challenges pertaining to thedistributed edge computing network that need to be ad-dressed for efficient and scalable implementation. Firstly,since several edge servers perform the distributed compu- • JS. Ng and WYB. Lim are with Alibaba Group and Alibaba-NTU JointResearch Institute, Nanyang Technological University, Singapore. • Z. Xiong is with Pillar of Information Systems Technology and Design,Singapore University of Technology Design. • D. Niyato is with School of Computer Science and Engineering, NanyangTechnological University, Singapore. • C. Leung is with The University of British Columbia and Joint NTU-UBCResearch Centre of Excellence in Active Living for the Elderly (LILY). • DI. Kim is with Sungkyunkwan University, South Korea. • J. Zhang is with School of Electrical, Computer and Energy Engineering,Arizona State University, USA. • Q. Yang is with Hong Kong University of Science and Technology, HongKong, China. tation tasks collaboratively, the communication costs can behigh due to the frequent exchange of intermediate results.Secondly, the response times vary across the edge serversdue to several factors such as imbalanced work allocation,contention of shared resources and network congestion [6],[7]. Thirdly, the confidentiality of the data may be com-promised as eavesdroppers may monitor data transmissionover wireless channels.Coded distributed computing (CDC) [8] has been pro-posed as an efficient method for distributed computationtasks at the edge of the network. In particular, coding tech-niques are used to design computation strategies that dividethe entire dataset and allocate subsets of data to the edgeservers for computations. In the distributed edge computingnetwork, one of the main challenges is the straggler effectswhere the task completion time is determined by the slowestedge server as the cloud server needs to wait for all edgeservers to return their results before it can reconstruct thefinal result. As a result, the latency of the distributed com-putation tasks can be high [9], [10]. By using CDC schemes ,instead of having to wait for all edge servers to completetheir computation tasks, the cloud server only needs to waitfor a subset of edge servers to return their results. Hence,CDC schemes can reduce computation latency by obviatingthe need to wait for the slower edge servers.However, incentives are essential for the edge serversto participate in or to complete their allocated CDC sub-tasks. To design an appropriate incentive mechanism, it is
1. CDC schemes do not only mitigate straggler effects, but can alsoreduce communication costs and ensure security in the distributed edgecomputing network. This paper focuses on CDC schemes that aim tomitigate straggler effects. a r X i v : . [ c s . G T ] F e b important to consider the unique characteristics of the CDCframework. Specifically, even though the edge servers areeach allocated a subset of the entire dataset for computa-tions, some of the edge servers’ computed results may notbe used to reconstruct the final result, e.g., due to straggling.These edge servers in turn do not receive any compensation.As a result, this may discourage the participation of certainedge servers. To address this challenge, we propose an all-pay auction to model the competition between the differentedge servers and at the same time, improve the participationof edge servers so as to elicit more CPU power for the CDCtasks.In distributed edge computing networks, the edgeservers may work together with various edge devices, byforming coalitions in order to complete their computationtasks. To model the cooperation between the edge serversand devices, we propose a hedonic coalition formation gamein which the edge devices decide which edge server to joinbased on their utility-maximizing objectives. In analogy topractical scenarios, the edge devices make decisions thatmaximize their utilities without taking into considerationthe effect of their decisions on other edge servers or devices.The main aim of this work is to develop an incentivemechanism for enabling efficient completion of CDC tasksfor IoT applications. Our key contributions are summarizedas follows:1) We highlight the importance of incentives in CDC,which is an issue ignored, but crucial toward eco-nomically sustainable distributed systems, by exist-ing works.2) We propose a two-level game theoretic approach toincentivize the edge servers to contribute their CPUpower for the CDC tasks.3) We formally show that the edge servers may im-prove their utilities by forming coalitions. We, there-fore, introduce a hedonic coalition formation gameto achieve a stable coalitional structure.4) We adopt an all-pay auction to model the competi-tion between the different edge servers (with theircoalitions of edge devices) which aim to win therewards offered by the cloud server and analyze thedifferent reward structures that affect the utility ofthe cloud server.5) We evaluate the performance of the proposedscheme. Simulation results show that the totalamount of CPU power allocated for the CDC tasksis higher under the proposed scheme as comparedto random CPU power allocation.The remainder of the paper is organized as follows.Section 2 highlights the related works. Section 3 presentsthe system model and problem formulation. Section 4 andSection 5 discuss the hedonic coalition formation gameand the design of an all-pay auction respectively. Section 6reports the simulation results and analysis of the proposedtwo-level game-theoretic approach. Section 7 concludes thepaper. ELATED W ORK
We discuss the recent studies related to three differentareas, i.e., (i) coded distributed computing, (ii) coalitional formation game, and (iii) auction design.
Given the emergence of big data which necessitatescomputation- and storage-intensive processing, large-scaledistributed systems have received significant attention fromboth the research and industrial communities. A numberof studies in the literature have focused on the minimiza-tion of communication load of the distributed computationtasks. Network coding in the context of distributed cachesystems has been a promising approach to increase networkthroughput and improve performance by jointly optimizingdata placement and delivery phases [11], [12].Recently, coding techniques have increasingly been usedin distributed computing networks. One of the active re-search areas is the minimization of the communicationload in the data shuffling phase through coded multicasttransmission as this phase accounts for a large proportion ofthe overall execution time [13]. There is a tradeoff betweencomputation load and communication load [14]. In orderto reduce the number of communication rounds, which issignificant for distributed iterative algorithms, [15] proposesa computing technique that jointly codes the computationat multiple iterations by leveraging on the storage andcomputation redundancy of the workers. The work in [16]considers the network topology of the distributed systemsin designing an efficient CDC scheme for practical imple-mentation. It relaxes the assumption that the physically-separated servers are connected to a single error-free com-mon communication bus.Apart from the studies that focus on the minimization ofcommunication load in the distributed computing networks,coding techniques are also used to alleviate the stragglers’delays that limit the performance as distributed computingsystems are scaled up. This is achieved by reducing therecovery threshold. Various CDC schemes are proposedfor different computation problems, e.g., matrix multipli-cation [17], [18], gradient descent [19], convolution [20],linear transform [21] and Fourier transform [22]. Insteadof ignoring the partial computations that are completed bythe stragglers, several studies such as [23] and [24] exploitthe work completed by the stragglers through sequentialprocessing and multi-message communication. In [25], thecomputation load is reduced by removing complex multipli-cation and division operations in the encoding and decodingphases.However, to the best of our knowledge, there are fewstudies that focus on the design of incentive mechanisms forCDC tasks. Given the interactions of autonomous and non-cooperative agents in the networks, an effective incentivemechanism design is an important step towards realizingthe scalable and efficient implementation of CDC schemesin distributed edge computing networks.
Due to the limited resources of a single device in completingthe allocated task individually, coalition formation gamesin computation offloading have been investigated. In [26],the fogs can cooperate with each other by sharing theirresources in order to offer better quality of service and experience for the users. A joint coalition-and-pricing baseddata offloading framework is proposed in [27] to maximizethe data throughput and determine the equilibrium pricesthat promote cooperation between devices and the edgeservers. Different from the generic coalitional formationgames where the utilities of the players depend on thecoalitional structures, the hedonic coalitions are formedbased on the individual preference of the players. As such,the utilities of the players depend solely on the members ofthe coalitions to which the worker belongs. In [28], a trust-based hedonic coalitional game is formulated to model theformation of trustworthy multi-cloud communities that areresilient to collusion attacks by malicious devices.It is often assumed that the resources of a coalition arededicated for a particular computation task. In practical sce-narios, each coalition may be required to complete multiplecomputation tasks. Hence, in order to incentivize the edgeservers to allocate more resources to complete the CDCtasks, we adopt an auction scheme.
The task of auction design for optimal allocation of re-sources and tasks is well-explored in the literature. In par-ticular, in crowdsensing applications where the usefulnessof the applications depends on the quantity and quality ofdata, auction theory is one of the important tools to achievemutual agreement between the crowd-sourcer and the users.Specifically, an all-pay auction is used to encourage thecontributions of users, e.g., data, that are used to solve acrowd-sourcing problem. In an all-pay auction, not all usersthat contribute to the task defined by the crowd-sourcer arerewarded. This is similar to the blockchain model illustratedin [29] where only the miner which successfully generates anew block is rewarded.In the literature, the design of the all-pay auctions con-siders different objectives and approaches. For example,some studies focus on the maximization of the quality of thecontributions [30] while others focus on the maximization ofthe sum of the contributions [31], [32]. The work in [33] stud-ies the total expected performance of asymmetric playersin competing for heterogeneous prizes under a complete-information setting. In contrast, the study of [34] considersan incomplete information setting where users do not knowhow other users value the reward offered by the crowd-sourcer. Besides, the crowd-sourcer maximizes its profit byrewarding the winner based on its contribution. Severalstudies such as [35] and [36] have analyzed the optimal prizestructures for crowdsensing platforms.However, the formation of coalitions in auctions hasseldom been considered. Here, we adopt a game-theoreticapproach to incentivize the edge servers to contribute theirCPU power for the CDC tasks.
YSTEM M ODEL AND P ROBLEM F ORMULATION
We consider a heterogeneous distributed edge computingnetwork as illustrated in Fig. 1. The system model consistsof a master, i.e., cloud server, and a set I = { , . . . , i, . . . , I } of I cluster heads, e.g., edge servers, that have different Cloud Server (Master) Edge Server (Cluster Head) Coalition Edge Device (Worker)Allocation of CPU Power for CDC subtask, 𝜏 i Reward, M k Splitting of Dataset U1 Hedonic Coalition FormationAll-pay Auction
U2 U3 L1L2
Reward Pool, p i CPU Power of Worker in coalition, z j Fig. 1:
System model consists of the cloud server (master), edge servers(cluster heads) and edge devices (workers). In the lower level, there aretwo steps: (L1) the workers in the coalitions allocate their CPU power,and (L2) the cluster heads offer a reward pool to the workers in thecoalitions. In the upper level, there are three steps: (U1) the master splitsthe dataset using polynomial codes, (U2) the cluster heads allocate CPUpower for the CDC subtasks, and (U3) the cluster heads are rewardedfor completing the allocated CDC subtasks. computational capabilities and belong to different serviceproviders. Moreover, there are J workers, e.g., edge de-vices, represented by the set J = { , . . . , j, . . . , J } , thatalso have different computational capabilities to facilitatein the computation tasks. In IoT networks, for example, theubiquity of the IoT devices as well as their on-board sensingand processing capabilities are leveraged to collect data formany innovative IoT applications. Given the large amountsof sensor data collected from different IoT devices, the mas-ter aims to perform the training of an AI model to completea user-defined data processing task. As the number of IoTdevices increases, so does the size of the dataset that themaster needs to handle. However, the master may not havesufficient resources, i.e., computation power, to handle thegrowing dataset. Instead, it may utilize the resources ofthe cluster heads to complete the computation tasks in adistributed manner. The cluster heads may cooperate withthe workers to increase their capabilities in completing thecomputation tasks. In particular, more CPU power can beallocated for the computation tasks. One of the main challenges in performing distributed com-putation tasks is the straggler effect. In order to reduce thecomputation latency of the distributed computation tasks,the master applies CDC schemes over the distributed edgecomputing network. Coding techniques such as polynomial
TABLE 1: System Model Parameters.
Parameter Description I Number of cluster heads J Number of workers K Recovery threshold ρ i Reward pool by cluster head v ( S i ) Value of coalition x S i j Utility of worker j in coalition S i σ Total amount of reward M k Size of reward ˜ A i , ˜ B i Allocated matrices to cluster heads ˜ C i Computed results by cluster heads π Expected utility of master z i CPU power of cluster head z j CPU power of worker µ ij Communication cost between worker j and clusterhead iκ Effective switch coefficient a i Total number of CPU cycles θ p Unit cost of computational energy θ c Unit cost of communication energy c i Communication energy of cluster head iτ i Allocated CPU power for the CDC subtasks α i Utility of cluster head u i Expected utility of cluster head u j ( S i ) Preference function of worker j in coalition S i v i Valuation of cluster head for total reward p ki Probability of winning the reward codes [17] can be used to mitigate straggler effects by reduc-ing the recovery threshold, i.e., the number of cluster headsthat need to submit their results for the master to reconstructthe final result. In order to perform coded distributed matrixmultiplication computations, i.e., C = A (cid:62) B where A and B are input matrices , A ∈ F s × rq and B ∈ F s × tq for integers s , r , and t and a sufficiently large finite field F q , there arefour important steps:1) Task Allocation:
Given that all cluster heads are ableto store up to m fraction of matrix A and n fractionof matrix B , the master divides the input matricesinto submatrices ˜ A i = f i ( A ) and ˜ B i = g i ( B ) ,where ˜ A i ∈ F s × rm q and ˜ B i ∈ F t × rn q respectively.Specifically, f and g represent the vectors of func-tions such that f = ( f , . . . , f i , . . . , f I ) and g =( g , . . . , g i , . . . , g I ) , respectively. Then, the masterdistributes the submatrices to the cluster heads overthe wireless channels for computations.2) Local Computation:
Each cluster head i is allocatedsubmatrices ˜ A i and ˜ B i by the master. Based on
2. The matrix multiplication may also involve more than two ma-trices. Our system model can be easily extended to solve the matrixmultiplication of more than two matrices. the allocated submatrices, the cluster heads performmatrix multiplication, i.e., ˜ A (cid:62) i ˜ B i , ∀ i ∈ I .3) Wireless Transmission:
Upon completion of the localcomputations, each cluster head transmits its com-puted results, i.e., ˜ C i = ˜ A (cid:62) i ˜ B i to the master overthe wireless communication channels.4) Reconstruction of Final Result:
By using coding tech-niques, the master is able to reconstruct the finalresult upon receiving K out of I computed resultsby using decoding functions. In other words, themaster does not need to wait for all I cluster headsto complete their allocated CDC subtasks. Note thatalthough there is no constraint on the decodingfunctions to be used, a low-complexity decodingfunction such as the Reed-Solomon decoding al-gorithm [37] ensures the efficiency of the overallmatrix multiplication computations.By using the polynomial codes [17], the optimum recov-ery threshold that can be achieved where each cluster headis able to store up to m of matrix A and n of matrix B isdefined as: K = mn. (1)The training of an AI model may involve various typesof distributed computation problems, e.g., matrix multipli-cation, stochastic gradient descent, convolution and Fouriertransform. Without loss of generality, we consider the dis-tributed matrix multiplication computations. Matrix multi-plication is an important operation underlying many dataanalytics applications, e.g., machine learning, scientific com-puting and graph processing [17].However, there needs to be an incentive for a clusterhead to be one of the K cluster heads to complete their localcomputations of CDC subtasks and return their computedresults to the master. In this paper, we focus our study on a two-level game-theoretic approach as follows: (i) in the lower level, weadopt a hedonic coalition formation game to investigate thecoalition formation of workers to facilitate the computationtasks of the cluster heads, and (ii) in the upper level, westudy the all-pay auction to encourage the cluster heads,given the coalitions of workers formed, to allocate moreCPU power for the CDC subtasks while maximizing theutility of the master. It is assumed that the cluster headsdo not consider forming coalitions among themselves sincethey are independent and competing service providers.
Given I cluster heads and J workers in the network, theformation of the coalitions is derived in the lower level. Inorder to encourage more workers to facilitate its compu-tation tasks, each cluster head offers a reward pool to thecoalition of workers. The reward pools for the cluster headsmaybe different depending on their available budgets. Thereward that each worker receives is a function of its pro-portion of CPU power contributed in the coalition, which,for example, can be measured from the computation latencyof that worker. On the one hand, workers are incentivized to join a cluster head that has a greater reward pool in thehope of receiving a higher reward. On the other hand, asmore workers join a cluster head, each worker will receive asmaller proportion of the reward pool as the pool needs tobe shared among more workers. In addition to the amountof rewards, the workers’ utilities are also affected by itscomputation and communication costs. Hence, the workersmake their decisions based on their utilities. Each worker j can choose to join any cluster head i ∈ I . Note thateach worker is only allowed to choose to facilitate thecomputation tasks of one of the cluster heads. In practice,worker j may be limited in the choice of cluster heads it canjoin, e.g., due to geographical location. The lower-level coalition formation game determines theamount of CPU power that each coalition has. The coalitionswith greater CPU power are more valuable to the master asthey are able to complete the CDC subtasks within a shorterperiod of time. Since a cluster head with its coalition ofworkers may need to work on several computation taskssimultaneously, they may not allocate all their CPU powerfor CDC subtasks. In order to incentivize the cluster headsto allocate more CPU power to complete the CDC subtasks,the master offers rewards to the cluster heads. Since com-puted results are required from only a subset of clusterheads, the cluster heads need to compete for the rewards.In particular, we explore an all-pay auction mechanismwhereby the cluster heads bid for the rewards. In this all-pay auction, although all cluster heads allocate CPU powerto perform the computations on the allocated dataset, only K cluster heads are rewarded. As such, the all-pay auction isdesigned such that the utility of the master is maximized byincentivizing the cluster heads to allocate more CPU powerfor the CDC subtasks.In traditional auctions such as first-price and second-price auctions, only the winners of the auctions pay. In con-trast, in all-pay auctions, regardless of whether the bidderswin or lose, they are required to pay to participate in theauction. In this all-pay auction, the bids of the edge serversare represented by their CPU power, i.e., the number ofCPU cycles, allocated by the edge servers to complete theCDC subtasks. In other words, the larger the CPU powerallocated, the higher the bid of the edge server.There are two advantages of an all-pay auction [38].Firstly, it reduces the probability of non-completion of allo-cated subtasks, thus allowing the cloud server to reconstructthe final result. This differs from traditional auctions inwhich the winners of the auctions can still choose not tocomplete their tasks and give up the reward that is promisedby the auctioneer (cloud server). As a result, the auctioneerneeds to conduct another round of auction. Secondly, itreduces the coordination cost between the auctioneer andthe bidders (edge servers). Specifically, in traditional auc-tions, the participants need to bid then contribute whereasin all-pay auctions, the bids of the participants are directlydetermined by their contributions. In other words, the par-ticipants do not need to bid explicitly in all-pay auctions.This is particularly useful for the development of a scalablenetwork since the communication overheads are reduced. In the lower level, the workers form coalitions to supportthe computation tasks of the cluster heads, increasing thecapabilities of the cluster heads to complete their compu-tation tasks by, for example, reducing computation latencyor increasing computation accuracy. Given the coalitions ofworkers formed, the cluster heads need to allocate CPUpower for their computation tasks. Without proper incentivemechanisms, the cluster heads may randomly allocate CPUpower for their computation tasks, which is not optimal asit does not maximize the utilities of the cluster heads. Inorder to incentivize the cluster heads to allocate CPU powerfor the CDC subtasks, an all-pay auction is proposed inthe upper level. The lower-level hedonic coalition formationgame helps to improve the utilities of the cluster heads by al-lowing them to allocate their equilibrium CPU power in theupper-level all-pay auction. Specifically, without formingcoalitions, the capabilities of the cluster heads are limitedby their own CPU power, thus not allowing them to allocatetheir equilibrium CPU power for the CDC subtasks, whichmay be greater than their own CPU power. As such, thecluster heads may not win the reward offered by the master.Therefore, the two-stage game theoretic approach ensuresthat the utilities of the cluster heads are maximized byallocating CPU power for the CDC subtasks.
OWER - LEVEL H EDONIC C OALITION F ORMA - TION
In this section, we formulate the problem of collaborativeexecution of computation tasks as a hedonic coalition forma-tion game. To form a coalition, each cluster head broadcastsits intention to form a coalition to all workers in the network.Each cluster head i offers a reward pool ρ i to the coalition ofworkers. To decide whether to join or leave a coalition, eachworker also compares its utility in the current coalition andthe utility of joining another coalition. If the utility of joininganother coalition is higher, the worker leaves the currentcoalition and joins another coalition, hence forming a newcoalitional structure. The coalitional structure is stable whenno worker has incentive to change its current coalition. We present the definitions for hedonic coalition formationformulation.
Definition 1.
A coalition of workers is denoted by S i ⊆ J where i is the index of the cluster head. In particular, workers in coalition S i facilitate the com-putation tasks of cluster head i , ∀ i ∈ I . Definition 2.
A partition or coalitional structure is a set ofcoalitions that spans all workers in J . The coalitional structure isrepresented by Π = { S , . . . , S i , . . . , S I } , where S i ∩ S i (cid:48) = ∅ for i (cid:54) = i (cid:48) , (cid:83) Ii =1 S i = J and I is the total number of coalitionsin coalitional structure Π [39]. J denotes the coalition of all workers, which is alsoknown as the grand coalition. The formation of a grandcoalition means that all workers facilitate the computationtasks of a single cluster head. A singleton coalition is a coalition that only contains a single worker where only aworker facilitates the computation tasks of a cluster head.Note that the total number of coalitions equals the numberof cluster heads in the network. If there is no worker thatis willing to facilitate the computation tasks of cluster head i , ∀ i ∈ I , the coalition S i associated with cluster head i isrepresented by an empty set, ∅ . Definition 3.
A coalitional structure Π ∗ = { S ∗ , . . . , S ∗ i , . . . , S ∗ I } is a stable coalitional structure if nocoalition S ∗ i ∈ Π has an incentive to change the currentcoalitonal structure Π by merging with another coalition S ∗ i (cid:48) , S ∗ i ∩ S ∗ i (cid:48) = ∅ for i (cid:54) = i (cid:48) , or splitting into smaller disjointcoalitions. The value of any coalition S i ∈ Π is the total amount ofCPU power of both the cluster head and workers in thecoalition. The value of coalition S i , which is denoted as v ( S i ) , is expressed as follows: v ( S i ) = (cid:88) j ∈ S i z j + z i , (2)where z j and z i are the amount of available CPU power ofworker j and cluster head i , respectively.Each cluster head i , ∀ i ∈ I , offers different amount ofreward pool, ρ i . Each worker j in coalition S i receives aproportion of the reward pool offered by the cluster head,for which the coalition S i provides support. The amount ofreward that each worker receives depends on its proportionof the CPU power in the coalition, which, for example, canbe measured from the worker’s computation latency. Thegreater the proportion of CPU power, the larger the amountof reward the worker receives. Specifically, the utility ofworker j in coalition S i , is denoted as follows: x S i j = z j (cid:80) j ∈ S i z j ρ i − δ j z j − µ ij , (3)where δ j is the unit cost of CPU power of worker j and µ ij is the communication cost for worker j to reach clusterhead i . In particular, the utility of worker j ∈ J dependsonly on the members of the coalition that it belongs to.Based on its own utility, each worker j ∈ J needs tobuild its own preference over all possible coalitions that itcan join, where each worker j compares the utilities of join-ing different coalitions. As such, the concept of preferencerelation is introduced to illustrate the preference of eachworker over all possible coalitions. Definition 4.
For any worker j ∈ J , a preference relation (cid:31) j isdefined as a complete, reflexive and transitive binary relation overthe set of all coalitions that worker j can possibly join [40]. The preference relation of worker j ∈ J can be ex-pressed as follows: S (cid:23) j S ⇐⇒ u j ( S ) ≥ u j ( S ) , (4)where S ⊆ J and S ⊆ J are two possible coalitionsthat worker j may join, u j ( S i ) is the preference functionfor any worker j ∈ J and for any coalition S i , ∀ i ∈ I .In particular, for any worker j ∈ J , given two coalitions S ⊆ J and S ⊆ J where j ∈ S and j ∈ S , S (cid:23) j S means that worker j prefers coalition S over coalition S , orat least worker j values both coalitions equally. In addition, its asymmetric counterpart, which is denoted as (cid:31) j , whenused in S (cid:31) j S indicates that worker j strictly preferscoalition S over coalition S . It is worth noting that thepreference relation, (cid:23) j is defined to allow the workers toquantify their preferences, which can be application-specific.The preference relation can be expressed as a function ofseveral parameters such as the payoffs of workers in joiningdifferent coalitions and the proportion of the contribution ofeach worker in the same coalition.The preference function of worker j in coalition S i whichis represented by u j ( S i ) is defined as follows: u j ( S i ) = (cid:40) x S i j , if S i / ∈ h ( j ) , −∞ , otherwise , (5)where x S i j is the utility of worker j in coalition S i definedin Equation (3) and h ( j ) is the history set of worker j thatcontains the list of coalitions that the worker j has previ-ously joined before the formation of the current coalitionalstructure Π . More specifically, the history set of worker j ∈ J , h ( j ) = { S i , . . . , S λi λ , . . . , S Λ i Λ } , where i λ ∈ I , j ∈ S i λ and Λ represents the total number of changesin coalitions formed by worker j . Each time when a newcoalition is formed, each worker j ∈ J updates its historyset h ( j ) by adding a new coalition S λi λ , where i λ ∈ I and j ∈ S i λ .As such, based on Equation (5), the preference of worker j ∈ J over the different coalitions is related to its utilitydefined in Equation (3).Given a set of workers J and a preference relation (cid:23) j for every worker j ∈ J , a hedonic coalition formation gameis formally defined as follows: Definition 5.
A hedonic coalition formation game is a coalitionalgame that is defined by ( J , (cid:31) ) where J and (cid:31) = {(cid:31) , · · · , (cid:31) j , · · · , (cid:31) J } represent the set of workers and the preference relationof each worker in J respectively. In addition, a hedonic coalitionformation game fulfils the two important requirements as follows: The payoff of any worker depends solely on the membersof the coalitions to which the worker belongs, and The coalition formed is a result of the preferences of theworkers over the set of possible coalitions.
In the hedonic coalition formation game, based on thepreference relation in Equation (3) and preference functionin Equation (5), the worker j ∈ J joins a new coalitionthat it has not visited before, and if and only if worker j achieves higher utility in the new coalition. Specifically, theformation of hedonic coalitions is based on switch rule,which determines whether the worker j ( ∀ j ∈ J ) decidesto leave or join a coalition. Definition 6. (Switch Rule) Given a coalitional structure
Π = { S , . . . , S i , . . . , S I } , a worker j decides to leave its currentcoalition S i and join another coalition S i (cid:48) ∈ Π , where i (cid:54) = i (cid:48) ,if and only if S i (cid:48) (cid:83) { j } (cid:31) j S i . As a result, { S i , S i (cid:48) } →{ S i \{ j } , S i (cid:48) (cid:83) { j }} . The switch rule in hedonic coalition formation gamesallows any worker j ∈ J to leave its current coalition S i and join another coalition S i (cid:48) ∈ Π , where i (cid:54) = i (cid:48) ,given that S i (cid:48) (cid:83) { j } is strictly preferred over S i based on the defined preference relation. This transforms the cur-rent coalitional structure Π into a new coalitional structure Π (cid:48) = Π \{ S i , S i (cid:48) } (cid:83) { S i \{ j } , S i (cid:48) (cid:83) { j }} . The adoption ofswitch rule in hedonic coalition formation games reflects theselfish behaviour of the workers since the workers decideto leave and join any coalition based on their preferencerelations, without taking into account the effect of theiractions on other workers. Proposition 1.
If worker j performs the switch rule for the λ -thtime, where it leaves its current coalition S λ − i λ − and forms a newcoalition S λi λ , where i λ − (cid:54) = i λ , the new coalition S λi λ cannot bethe same as any coalition formed in the history set, h ( j ) . In otherwords, before the update of the history set for the λ -th time, thenew coalition is not in the history set, i.e., S λi λ / ∈ h ( j ) .Proof. Suppose that there are two coalitions S λ − i λ − and S λi λ such that S λ − i λ − = S λi λ , where coalition S λ − i λ − is foundin the history set, h ( j ) . Based on Equation (3), the utilityof worker j in either of the coalition is the same, i.e., x S λ − iλ − j = x S λiλ j . According to the definition of switch rule inDefinition (6), worker j only performs the switch operationif and only if the new coalition is strictly preferred over anyof the previous coalitions. In other words, switch operationis only performed if and only if x S λiλ j > x S λ − iλ − j . Since thenewly formed coalition S λi λ does not fulfil the condition ofswitch rule, the switch operation is not performed. Morespecifically, the newly formed coalition S λi λ cannot be thesame as any coalition S λi λ in the history set, h ( j ) .In the formation of hedonic coalitions, there exists astable coalitional structure. There are two types of stabilitiesof coalitional structure, i.e., Nash-stability and individual-stability [40]. • Nash-stability:
A coalitional structure
Π = { S , . . . , S i , . . . , S I } is Nash-stable if no worker j ∈ J has incentive to leave its current coalition S i and join another coalition S i (cid:48) where i (cid:54) = i (cid:48) , i.e., S i (cid:31) S i (cid:48) (cid:83) { j } , ∀ i (cid:48) ∈ I . In other words, no worker isable to increase its utility by performing switch ruleto change its current coalition. • Individual-stability:
A coalitional structure
Π = { S , . . . , S i , . . . , S I } is individually-stable if theredoes not exist such that (i) a worker j , ∀ j ∈ J in itscurrent coalition strictly prefers any other coalition,i.e., S i (cid:48) (cid:83) { j } (cid:31) j S i , ∀ i ∈ I , and (ii) the formationof a new coalition does not reduce the utilities of themembers of the new coalition, i.e., S i (cid:48) (cid:83) { j } (cid:31) j (cid:48) S i (cid:48) , j (cid:54) = j (cid:48) , ∀ j (cid:48) ∈ S i (cid:48) .Note that when a coalitional structure is Nash-stable, it isalso individually-stable [40] since Nash-stability is a subsetof individual-stability. Proposition 2.
The final partition Π ∗ = { S ∗ , . . . , S ∗ i , . . . , S ∗ I } is a Nash-stable and individually-stable coalitional structure.Proof. Given any current coalitional structure Π curr = { S , . . . , S i , . . . , S I } , where S i ∩ S i (cid:48) = ∅ for i (cid:54) = i (cid:48) , (cid:83) Ii =1 S i = J , switch operations are performed for any worker j ∈ J toeither leave or join a coalition. The current partition Π curr is updated when the utility of any worker j ∈ J in anycoalition S i is higher by leaving its current coalition S i andjoining another coalition S i (cid:48) ∈ Π curr , for i (cid:54) = i (cid:48) . Accordingto Equation (5) which defines the preference function ofeach worker, the worker does not visit the coalitions thatare contained in its history set. Therefore, the hedonic coali-tional formation algorithm generates a sequence of coali-tional structures where each coalitional structure has notbeen visited before. The algorithm will eventually terminateat a final coalitional structure Π ∗ = { S ∗ , . . . , S ∗ i , . . . , S ∗ I } ,where there is no incentive for any worker to change itscurrent coalition. In other words, the utility of each worker j , ∀ j ∈ J , is maximized given the final coalitional structure Π ∗ .Suppose we assume that the final coalitional structure Π ∗ is not Nash-stable. This implies that there is a worker j ∈ J that has incentive to change its current coalition.As a result, by leaving its current coalition and joininganother coalition, the coalitional structure Π ∗ is updatedbased on the switch rule defined in Definition 6. Thus,the coalitional structure is not final, which does not alignwith our assumption that the final coalitional structure isnot Nash-stable. Therefore, the final coalitional structure Π ∗ must be Nash-stable. Since final coalitional structure Π ∗ isNash-stable, it is also individually-stable. The algorithm for the hedonic coalition formation is pre-sented in Algorithm 1.The hedonic coalition formation game is based on theswitch rule defined in Definition 6. The switch operationis illustrated from the perspective of worker j ∈ J . Theworker j decides to leave its current coalition S i and join an-other coalition S i (cid:48) where i (cid:54) = i (cid:48) and S i (cid:48) ⊆ Π curr if and onlyif the worker j achieves higher utility by joining coalition S i (cid:48) than that of the current coalition S i . The worker j firstcompute its utility in the current coalition S i , x S i j (line 8).Given the current coalitional structure Π curr , the worker j evaluates the other coalitions S i (cid:48) , for i (cid:54) = i (cid:48) that it couldpossibly join (line 9-10). Specifically, the worker j computesthe utility that it achieves if it joins another coalition S i (cid:48) , x S i (cid:48) j (line 10). If the utility of worker j for joining coalition S i (cid:48) ishigher than that of the current coalition S i and the coalition S i (cid:48) is not found in the history set of worker j , h ( j ) , theworker j performs switch operation (line 11-16). In particu-lar, the worker j first updates its history set by adding thecurrent coalition S i into h ( j ) (line 12). The worker j leavesits current coalition S i (line 13) and joins the new coalition S i (cid:48) (line 14). Then, given the new coalition S i (cid:48) , the currentcoalition and coalitional structure are updated (line 15-16).On the other hand, if the utility of worker for joining coali-tion S i (cid:48) is lower than that of the current coalition S i or thenew coalition S i (cid:48) has been visited before, the worker j doesnot leave its current coalition S i , thus there is no change inthe coalitional structure. For the next iteration, the worker j will consider to join other possible coalition S i (cid:48) ∈ Π curr ,for i (cid:54) = i (cid:48) . The process is repeated for all workers j ∈ J .The switch mechanism terminates when there is no morechange to the current coalitional structure, Π curr . In otherwords, there is no worker j ( ∀ j ∈ J ) that is able to achieve Algorithm 1
Algorithm for Hedonic Coalition formation ofWorkers using Switch Rule.
Input:
Set of workers, J = { , . . . , j, . . . , J } , set of clusterheads, I = { , . . . , i, . . . , I } Output:
Final coalitional structure Π ∗ = { S ∗ , . . . , S ∗ i , . . . , S ∗ I } Π ∗ = ∅ Initialize history set for all workers, i.e., h ( j ) = ∅ , ∀ j ∈J Given J workers, initialize a coalitional structure Π curr where workers are randomly allocated to the I coalitions Switch Rule: while Π curr (cid:54) = Π ∗ do Update the final coalitional structure such that Π ∗ =Π curr for each worker j ∈ J (worker j is in coalition S i ∈ coalitional structure Π curr ) do Compute x S i j for each possible coalition S i (cid:48) ∈ Π curr , i (cid:54) = i (cid:48) do Compute x S i (cid:48) j if x S i (cid:48) j > x S i j and S i (cid:48) / ∈ h ( j ) then Worker j updates its history set, h ( j ) by addingthe current coalition S i into h ( j ) Worker j leaves its current coalition, S i = S i \{ j } Worker j joins the new coalition that increasesits utility, S i (cid:48) = S i (cid:48) (cid:83) { j } Update current coalition of worker j , S i ← S i (cid:48) Update current coalitional structure Π curr ← Π curr \{ S i , S i (cid:48) } (cid:83) { S i \{ j } , S i (cid:48) (cid:83) { j }} end if end for end for end while return Final coalitional structure Π ∗ = { S ∗ , . . . , S ∗ i , . . . , S ∗ I } that is Nash-stablehigher utility by leaving its current coalition and join anyother coalition in the current coalition structure, Π curr . Atthe end of the switch mechanism, the algorithm returns thefinal coalitional structure Π ∗ = { S ∗ , . . . , S ∗ i , . . . , S ∗ I } thatis Nash-stable (line 20). Consequently, the total amount ofCPU power that each coalition can be computed, in whichthe competition between the different cluster heads arediscussed in the next section. PPER - LEVEL A LL - PAY A UCTION
Since the cluster heads, given their coalitions of workersformed, may have several computation tasks to complete,they only allocate a fraction of their CPU power to the CDCtasks. Hence, in order to incentivize the cluster heads toallocate more CPU power for the allocated CDC subtasks,we present the design of an all-pay auction in this section.In this all-pay auction, the master is the auctioneer whereasthe cluster heads are the bidders. The bid of a cluster headis represented by the CPU power that it allocates for itsCDC subtask, which, for example, can be measured from the computation latency incurred for the CDC subtask. All I cluster heads, i.e., bidders, pay their bids regardless ofwhether they win or lose the auction. We first discuss theutilities of both the master and the cluster heads. Then, wepresent the design of an all-pay auction. Given that the master only needs the computed resultsfrom K cluster heads to reconstruct the final result, themaster offers K rewards, represented by the set K = { , . . . , k, . . . , K } where K ≤ I . Specifically, there are K rewards for which I cluster heads compete. The effect ofdifferent reward structures is discussed later in details inSection 5.4. Since only K rewards are offered, I − K clusterheads do not receive any reward from the master, eventhough they perform the matrix multiplication computa-tions given the allocated submatrices. The all-pay auctionis designed such that the cluster heads are incentivized toallocate their CPU power, even if there is a possibility thatthey may not win any reward.The size of reward k is represented by M k . The clusterhead that allocates larger CPU power is offered larger re-ward. In particular, the cluster head that allocates the largestamount of CPU power receives a reward of M , the clusterhead with the second largest allocation receives reward M and the cluster head with the k -th largest allocation of CPUpower is offered reward M k . If two or more cluster headsallocate the same amount of CPU power to perform theCDC tasks, ties will be randomly broken. In other words,if both cluster heads are ranked k , one is ranked k andthe other is ranked k + 1 . Hence, without loss of generality, M ≥ M ≥ · · · ≥ M K > . The total amount of rewardoffered by the master is denoted by σ , i.e., σ = (cid:80) Kk =1 M k .The master broadcasts the information of size of total re-ward and the structure of rewards to the workers. The aimof the master is to share the entire fixed reward to maximizethe CPU power allocated by the cluster heads.As such, the expected utility of the master, π is expressedas follows: π = E [ φ ( τ I + τ I + · · · + τ K : I ) − σ ] , (6)where φ is the unit worth of CPU power to the masterand τ k : I represents the order statistics of the cluster head’sCPU power allocation. Specifically, τ I and τ k : I denote thehighest and k -th highest CPU power allocation respectivelyamong I cluster heads. To perform the local computations on the allocated CDCsubtask, each cluster head i consumes computational en-ergy, e i , which is defined as: e i = κa i ( τ i ) , (7)where κ is the effective switch coefficient that dependson the chip architecture [41], a i is the total number ofCPU cycles required to complete the allocated computationsubtask and τ i is the CPU power allocated by cluster head i for the CDC subtask. In other words, v ( S i ) − τ i is theamount of CPU power allocated by cluster head i for other computation tasks. By using the polynomial codes, the com-putation task is evenly partitioned and distributed amongall cluster heads. As a result, the total number of CPU cyclesthat are needed to complete the allocated computation tasksis the same for all cluster heads, i.e., a i = a, ∀ i ∈ I .The unit cost of computational energy incurred by clusterhead i , ∀ i ∈ I , is denoted by θ p , where the unit costof computational energy is the same for all cluster heads.Besides, each cluster head i also requires communicationenergy c i to communicate with the master. Similarly, theunit cost of communication energy is the same for all clusterhead where the unit cost of communication energy incurredby cluster head i , ∀ i ∈ I , is denoted by θ c .Each cluster head i has a valuation v i for the totalreward σ . For example, in practical scenarios, the valuationsfor the total reward can be determined by how much thecluster heads can benefit from the reward, which is a userpreference parameter. In particular, the cluster heads valuethe reward more if they need the reward for some importantpurposes, e.g. upgrading of their hardware components.The cluster heads’ valuations, v i , ∀ i ∈ I , are independentlydrawn from v i ∈ [ v, ¯ v ] such that v and ¯ v are strictly positivegiven F ( v ) , where F ( v ) is the cumulative distribution func-tion (CDF) of v . The total cost of cluster head i is representedby θ p e i + θ c c i . As a result, the utility of cluster head i forwinning reward M k , ∀ k ∈ K , is expressed as: α i = (cid:40) v i M k − θ p e i − θ c c i , if cluster head i wins M k , − θe i , otherwise . (8) Each cluster head i knows its own valuation, v i but doesnot know the valuation of any other cluster head, i (cid:48) (cid:54) = i .This establishes a one-dimensional incomplete informationsetting. In addition, if each cluster head has different unitcosts of computational and communication energy whichare only known to itself, we consider the three-dimensionalincomplete information setting. The dimension of privateinformation can be reduced following the procedure in [31].In this work, we consider a one-dimensional incompleteinformation setting where the unit costs of computationaland communication energy are the same for all cluster headsbut the cluster heads’ valuations are heterogeneous andprivate.Given the utility of cluster head i , α i in Equation (8), theobjective of cluster head i to maximize its expected utility, u i , is defined as follows: max τ i u i = v i K (cid:88) k =1 p ki M k − θ p κa ( τ i ) − θ c c i , (9)where p ki is the winning probability of reward M k by clusterhead i .Although the cluster head does not know exactly thevaluations of other cluster heads, it knows the distributionof the other cluster heads’ valuations based on past interac-tions, which is a common knowledge to all cluster heads andthe master. In our model, we consider that the valuationsof all cluster heads are drawn from the same distribution, which constitutes a symmetric Bayesian game where theprior is the distribution of the cluster heads’ valuations. Definition 7. [34] A pure-strategy Bayesian Nash equilibriumis a strategy profile τ ∗ = ( τ ∗ , . . . , τ ∗ i , . . . , τ ∗ I ) that satisfies u i ( τ ∗ i , τ ∗− i ) ≥ u i ( τ i , τ ∗− i ) , ∀ i ∈ I . The subscript − i represents the index of other clus-ter heads other than cluster head i . Specifically, τ ∗− i =( τ ∗ , τ ∗ , . . . , τ ∗ i − , τ ∗ i +1 , . . . , τ ∗ I ) represents the equilibriumCPU power allocations of all other cluster heads other thanCPU power allocation of cluster head i . At the BayesianNash equilibrium, given the belief of cluster head i , ∀ i ∈ I about the valuations and that the CPU power allocated byother cluster heads, i (cid:48) where i (cid:54) = i (cid:48) are at equilibrium, τ ∗ i (cid:48) ,cluster head i aims to maximize its expected utility. Proposition 3.
Under incomplete information setting, the all-pay auction admits a pure-strategy Bayesian Nash equilibriumthat is strictly monotonic where the bid of a cluster head strictlyincreases in its valuation.
Since the equilibrium CPU power allocation of clusterhead i , which is represented by τ ∗ i , is a strictly monoton-ically increasing function of its valuation v ∗ i , we expressthe equilibrium strategy of cluster head i as a functionrepresented by β ( · ) , i.e., τ ∗ i = β i ( v i ) . Given the strict mono-tonicity, the inverse function also exists where v i ( · ) = β − i ( · ) and it is an increasing function. Due to the incomplete infor-mation setting, the objective of cluster head i to maximize itsexpected utility in Equation (9) can be expressed as follows: max τ i u i = v i K (cid:88) k =1 p ki ( τ i , β i (cid:48) ( v i (cid:48) )) M k − c ( β ( v i )) , (10)where the cost of cluster head i is represented by thefunction c ( · ) = θ p κa ( β ( v i )) + θ c c i .Since the cluster heads are symmetric, i.e., the valuationsof cluster heads are drawn from the same distribution,the symmetric equilibrium strategy for each cluster head i , ∀ i ∈ I can be derived. We first assume that there are I rewards, where M ≥ M ≥ · · · ≥ M K > M K +1 = M K +2 = · · · = M I = 0 . The valuations of the cluster heads, v , . . . , v i , . . . , v I are ranked and represented by its orderstatistics, which are expressed as v I ≥ v I ≥ · · · ≥ v I : I .In particular, v k : I represents the k -th highest valuationamong the I valuations which are drawn from a commondistribution F ( v ) . Given the order statistics of the clusterheads’ valuations, ∀ i ∈ I , the corresponding cumulativedistribution function and probability density function arerepresented by F k : I and f k : I respectively. Specifically, thecumulative distribution function F k : I ( v ) for the k -th orderstatistics in sample of size I is expressed as follows: F k : I ( v ) = k − (cid:88) r =0 F ( v ) I − r [1 − F ( v )] r . (11)The corresponding probability density function f k : I ( v ) for k -th order statistics in sample of size I is expressed asfollows: f k : I ( v ) = I !( k − I − k )! F ( v ) ( I − k ) [1 − F ( v )] k − f ( v ) . (12) Similarly, when dealing with the valuations of all clusterheads, other than that of cluster head i , the order statisticis represented by v k : I − , which represents the k -th highestvaluation among the I − valuations. The correspondingcumulative distribution function and probability densityfunction are represented by F k : I − and f k : I − respectively.Given that other cluster heads i (cid:48) , where i (cid:48) (cid:54) = i , fol-low a symmetric, increasing and differentiable equilibriumstrategy β ( · ) , cluster head i will never choose to allocatea CPU power greater than the equilibrium strategy giventhe highest valuation. In other words, cluster head i willnever allocate τ i > β (¯ v ) . Besides, the optimal strategy of thecluster head with lowest valuation v is not to allocate anyCPU power. On one hand, when the number of rewardsoffered is smaller than the number of cluster heads, i.e., K < I , the cluster head with lowest valuation v will notwin any reward. On the other hand, when the numberof rewards offered is larger than or equal the number ofcluster heads, i.e., K ≥ I , the cluster head with lowestvaluation v will win a reward without allocating any CPUpower. Hence, u i ( v ) = 0 . With this, the expected utility ofcluster head i with valuation v i and CPU power allocation τ i = β ( v i ) is expressed as follows: u i = v i I (cid:88) k =1 [ F k : I − ( v i ) − F k − I − ( v i )] M k − c ( β ( v i )) , (13)since M k +1 = · · · = M I − = M I = 0 , F I − ( τ i ) ≡ and F I : I − ( τ i ) ≡ .By differentiating Equation (13) with respect to the vari-able w i and equating the result to zero, we obtain thefollowing: v i I (cid:88) k =1 [ f k : I − ( v i ) − f k − I − ( v i )] M k − c (cid:48) ( β ( v i )) β (cid:48) ( v i ) . (14)When maximized, the marginal value of the reward isequivalent to the marginal cost of the CPU power. Sincewe have the differentiated function c (cid:48) ( · ) , the function c ( · ) can be found by using the integral of Equation (14). Atequilibrium, when the expected utility of cluster head i , ∀ i ∈ I , is maximized, we have the following: c ( β ( v i )) = I (cid:88) k =1 M k (cid:90) v i v v i [ f k : I − ( v i ) − f k − I − ( v i )] dv i = I − (cid:88) k =1 ( M k − M k +1 ) (cid:90) v i v v i f k : I − ( v i ) dv i . (15)Thus the equilibrium strategy for cluster head i withvaluation v i , ∀ i ∈ I , is expressed as: τ ∗ i = β ( v i ) = c − (cid:32) I − (cid:88) k =1 ( M k − M k +1 ) (cid:90) v i v v i f k : I − ( v i ) dv i (cid:33) . (16)Given the equilibrium strategy of cluster head i , ∀ i ∈ I ,the master aims to maximize its expected utility, π . Byusing the polynomial codes, the master is able to recon-struct the final result by using the computed results from K cluster heads. Since the master shares the fixed reward σ completely, the maximization problem in Equation (6) is equivalent to maximizing the allocation of CPU power,which is expressed as follows: π = E [ β ( v I ) + β ( v I ) + · · · + β ( v K : I )]= K (cid:88) i =1 (cid:90) ¯ vv β ( v ) dF i : I ( v )= K (cid:90) ¯ vv β ( v ) dF ( v )= K (cid:90) ¯ vv c − (cid:32) I − (cid:88) k =1 ( M k − M k +1 ) (cid:90) vv vf k : I − ( v ) dv (cid:33) dF ( v )= K (cid:90) ¯ vv c − ( I (cid:88) k =1 M k (cid:90) vv v [ f k : I − ( v ) − f k − I − ( v ) dv ]) dF ( v ) . (17)Since the equilibrium strategy of cluster head i , ∀ i ∈ I , isaffected by the reward structure, the master needs to deter-mine the structure of the rewards such that it maximizesthe CPU power allocation of the cluster heads, therebymaximizing its own utility, π . Given that the master shares the total amount of the re-ward, σ , the design of the optimal reward sequence is im-portant to maximize the CPU power allocation of the clusterheads since the equilibrium strategies of the cluster headsdepend on the differences between consecutive rewards.The master needs to first decide whether to allocate thetotal amount of reward, σ to only one winner, i.e., winner-take-all reward structure, or to split the reward into severalsmaller rewards. Proposition 4.
Given that the cost functions are convex, it isnot optimal to offer only one reward where M = σ and M = · · · = M K = · · · = M I = 0 since ∂π∂M k − − ∂π∂M k < , for k = 2 , . . . , I . In particular, if ∂π∂M − ∂π∂M < , it is not optimalto offer only a reward.Proof. Following the procedure in [31], we show that it isnot optimal to offer a single reward given the cost functionsof the cluster heads are convex. ∂π∂M − ∂π∂M = K (cid:90) ¯ vv ( c − ) (cid:48) ( (cid:90) vv vf I − ( v ) dv ) × (cid:40) (cid:90) vv vf I − ( v ) dv − (cid:90) vv vf I − ( v ) dv (cid:41) dF ( v ) .∂∂v (cid:32) (cid:90) vv vf I − ( v ) dv − (cid:90) vv vf I − ( v ) dv (cid:33) = v { I − I − F ( v ) I − f ( v ) − ( I − I − F ( v ) I − × [1 − F ( v )] f ( v ) } = vf ( v ) F ( v ) I − (2( I − F ( v ) − ( I − I − − F ( v )]) . Let x = F ( v ) , the expression above is simplified to: ∂∂v (cid:32) (cid:90) vv vf I − ( v ) dv − (cid:90) vv vf I − ( v ) dv (cid:33) = vf ( v ) x I − (2( I − x − ( I − I − − x ]) . When x = 0 , ∂∂v ( · ) = vf ( v ) x I − [ − ( I − I − < .When x = 1 , ∂∂v ( · ) = vf ( v ) x I − I − > . As a result,there is ˆ x = F (ˆ v ) with ˆ v ∈ ( v, ¯ v ) such that ∂∂v (cid:32) (cid:90) vv vf I − ( v ) dv − (cid:90) vv vf I − ( v ) dv (cid:33) > , if and only if v > ˆ v . As such, this implies that there is v ∗ ∈ ( v, ¯ v ) such that (cid:90) vv vf I − ( v ) dv − (cid:90) vv vf I − ( v ) dv > , if and only if v > ˆ v ∗ . Given that (cid:90) ¯ vv (cid:90) vv vf I − ( v ) dv − (cid:90) vv vf I − ( v ) dvdF ( v ) > , and ( c − ) (cid:48) ( · ) < and ( c − ) (cid:48)(cid:48) ( · ) ≥ due to the convexity ofthe cost function, ∂π∂M − ∂π∂M < . Hence, it is not optimal toallocate only one reward to the cluster head which allocatesthe largest amount of CPU power, where M = σ and M = · · · = M I = 0 . Note that the similar procedure can be usedto proof for the general case of ∂π∂M − ∂π∂M k < for k =2 , . . . , I − .Since the winner-take-all reward structure is not optimal,the master is better off offering multiple rewards. Given that K rewards are offered, the master can consider several re-ward sequences such as (i) homogeneous reward sequence,(ii) arithmetic reward sequence and (iii) geometric rewardsequence. Specifically, the reward sequence is expressed asfollows: • Homogeneous reward sequence: M k = M k +1 , • Arithmetic reward sequence: M k − M k +1 = γ , γ > , • Geometric reward sequence: M k +1 = ηM k , ≤ η ≤ ,where γ and η are constants. IMULATION R ESULTS
In this section, we evaluate the two-level game theoreticapproach. We first analyze the hedonic coalition formationgame that maximizes the utilities of the workers, followedby the all-pay auction. In particular, we evaluate the be-haviour of the cluster heads in allocating their CPU powerfor the CDC subtasks. Table 2 summarizes the simulationparameter values.We consider a nomalized total amount of reward σ of ,i.e., σ = (cid:80) Kk =1 M k = 1 . We also set m = n = 2 (see “TaskAllocation” step in Section 3) and assume that the clusterheads are able to store equal size of the input matrices, A and B such that m = n = 2 . In the network, there are cluster heads and workers withdifferent CPU powers. We consider the hedonic coalitionformation game among the cluster heads and workers. Theobjective of each worker is to maximize its own utility,which depends solely on the members of the coalitionit belongs to. In particular, the utility of each worker isaffected by its proportion of CPU power in the coalition. TABLE 2: System Simulation Parameter Values. Parameter Values
CPU power of worker j , z j [100 , CPU power of cluster head i , z i [750 , Communication cost between worker j and clusterhead i , µ ij θ p θ c i , c i Effective switch coefficient, κ [42] − Total number of CPU cycles required, a × Valuation of cluster head i , v i ∼ U [0 , TABLE 3:
Simulation Parame-ter Values of the Cluster Heads.ClusterHead(CH)ID CPUPower(W) RewardCH 1 750 100CH 2 1000 90CH 3 1250 80CH 4 1500 70CH 5 1750 60
TABLE 4:
Simulation Parame-ter Values of the Workers.WorkerID CPUPower(W) UnitCostWorker 1 100 0.01Worker 2 150 0.02Worker 3 200 0.03Worker 4 250 0.04Worker 5 300 0.05Worker 6 350 0.06Worker 7 400 0.07Worker 8 450 0.08
The simulation parameter values of the cluster heads andthe workers are listed in Tables 3 and 4 respectively.The hedonic coalition formation game allows the work-ers to decide which cluster head to join. In order to decidewhether to stay in or leave a coalition, the workers adopt theswitch rule. Figure 2 illustrates the mechanism of the switchoperations. Initially, the workers are randomly assignedto the cluster heads. Each time the coalitional structurechanges, each worker evaluates its utility by comparing theutility achieved in the current coalition against the utilitygain from joining other possible coalitions. As a result, eachworker may perform more than one switch operation. Asan example, worker 1 achieves a utility of by joiningcluster head 2. As workers 3 and 5 join the coalition insupporting cluster head 2, the utility of worker 1 decreasesto . Worker 1 then decides to leave cluster head 2 andjoins worker 7 to support cluster head 1 as it gains a higherutility of . However, when worker 2 joins the coalition tosupport cluster head 1, worker 1’s utility decreases to . .As such, worker 1 decides to perform a switch operationagain where it joins worker 6 in supporting cluster head 3,achieving a utility of . .From Fig. 3, we observe that as the amount of rewardpool offered by a cluster head increases, the total amountof CPU power of the workers in the coalition increases. Forexample, cluster head 1 offers a reward of 100 and forms acoalition with worker 2 and worker 7 having CPU powersof W and
W respectively. {3, 5} {1} {4} {6, 7, 8} {2}{5} {1, 3} {4} {6, 7, 8} {2}{5, 6, 7} {1, 3} {4} {8} {2}{7} {1, 3, 5} {4, 6} {8} {2}{7, 1} {3, 5} {6} {8} {2, 4}{7, 1, 2} {3, 5} {6} {8} {4} Cluster Head 1 Cluster Head 2 Cluster Head 3 Cluster Head 4 Cluster Head 5
Worker 3 switchesInitializationWorkers 6 and 7 switchWorkers 5 and 6 switchWorkers 1 and 4 switchesWorker 2 switches S w i t c h O p e r a t i o n s {7, 2} {3, 5} {6, 1} {8} {4} Worker 1 switchesFINAL
Fig. 2: Switch operations of the hedonic coalition formationgame.
60 65 70 75 80 85 90 95 100Reward Pool of Cluster Head i C P U P o w e r b y W o r k e r s i n C o a li t i o n S i ( W ) i=5 i=4 i=3 i=2 i=1 Fig. 3: CPU power by workers in coalition S i vs the rewardpool offered by cluster head i . In the simulations, we consider a uniform distribution of thecluster heads’ valuation for the rewards, where v i ∈ [0 , which are independently drawn from F ( v ) = v . FromFigs. 4-9, it can be observed that the cluster head’s CPUpower allocation increases monotonically with its valuation.Specifically, the higher the valuation of the cluster head forthe rewards, the larger the amount of CPU power allocatedfor the CDC subtask. Since the cluster heads are symmetricwhere their valuations are drawn from the same distribu-tion, the cluster heads with the same valuation contributethe same amount of CPU power. Based on the different reward structure adopted by the mas-ter, the cluster heads allocate their CPU power accordingly.Figure 4 shows that when there are 5 workers and only one reward is offered to the cluster head that allocates the largestamount of CPU power, the cluster head with the highestvaluation of 1, i.e., v i = 1 , is only willing to contribute . W of CPU power. However, when the master offersmultiple rewards, the cluster head with the same valuationof 1 is willing to contribute as high as W, W and
W as shown in Fig. 6, Fig. 7 and Fig. 8 respectively.With more rewards, the cluster heads have higher chance ofwinning one of the rewards. Hence, to incentivize the clusterheads to allocate more CPU power for the CDC subtasks, themaster is better off offering multiple rewards than a singlereward.
The master needs to decide between homogeneous andheterogeneous reward allocation. Homogeneous rewardsmeans the total amount of reward is split equally amongthe winning cluster heads whereas heterogeneous rewardsare allocated based on the rank of the cluster heads wherethe amount of reward offered to the cluster head decreasesas its rank increases. • Homogeneous Reward Allocation:
We observe sim-ilar trends in both homogeneous and heterogeneousreward allocation. Specifically, the cluster heads withlower valuations allocate more CPU power whenthere are fewer cluster heads in the network whereasthe cluster heads with higher valuations allocatemore CPU power when there are more cluster headsin the network. Generally, the cluster heads of bothlow and high valuations allocate more CPU powerwhen homogeneous rewards are allocated. Figure 5shows that when there are 10 cluster heads in thenetwork, a cluster head with valuation of . allo-cates W, which is higher than W, Wand
W in Fig. 6, Fig. 7 and Fig. 8 respectively.Similarly, in a network with 10 cluster heads, acluster head with valuation of . allocates . Wwhen homogeneous rewards are allocated, which isalso higher than . W, . W and . W when thedifferences between the consecutive rewards are afactor of 0.8, a constant of 0.05 and 0.1 respectively. • Heterogeneous Reward Allocation:
Figure 6, Fig. 7and Fig. 8 show the allocation of CPU power by thecluster heads under arithmetic and geometric rewardsequences. When the difference between the consec-utive rewards is smaller, the cluster heads are willingto allocate more CPU power. For example, when thedifference between the consecutive rewards is 0.05,i.e., M k − M k +1 = 0 . , k = 1 , , . . . , K − , thecluster head with valuation of 1 allocates Wwhen there are 15 cluster heads competing for 4rewards. However, under the same setting of 15cluster heads competing for 4 rewards, the clusterhead with valuation of 1 is only willing to allocate
W when the difference between the consecutiverewards is 0.1.
Apart from the different reward structures, the cluster headsalso behave differently when the system parameter values, v i C l u s t e r H e a d ' s C P U P o w e r A ll o c a t i o n , i Number of Cluster Heads, I =5Number of Cluster Heads, I =10Number of Cluster Heads, I =15 Fig. 4: Only one reward is offered tothe worker with the largest CPU powerallocation. v i C l u s t e r H e a d ' s C P U P o w e r A ll o c a t i o n , i Number of Cluster Heads, I =5Number of Cluster Heads, I =10Number of Cluster Heads, I =15 Fig. 5: Homogeneous rewards, i.e., thedifference between consecutive rewardamounts is . v i C l u s t e r H e a d ' s C P U P o w e r A ll o c a t i o n , i Number of Cluster Heads, I =5Number of Cluster Heads, I =10Number of Cluster Heads, I =15 Fig. 6: The difference between consec-utive reward amounts is by a factor of0.8, M k +1 = 0 . M k . v i C l u s t e r H e a d ' s C P U P o w e r A ll o c a t i o n , i Number of Cluster Heads, I =5Number of Cluster Heads, I =10Number of Cluster Heads, I =15 Fig. 7: The difference M k − M k +1 be-tween consecutive reward amounts is0.05. v i C l u s t e r H e a d ' s C P U P o w e r A ll o c a t i o n , i Number of Cluster Heads, I =5Number of Cluster Heads, I =10Number of Cluster Heads, I =15 Fig. 8: The difference M k − M k +1 be-tween consecutive reward amounts is0.1. v i C l u s t e r H e a d ' s C P U P o w e r A ll o c a t i o n , i K=3K=4K=5
Fig. 9: Different number K of rewards,the difference between the consecutiverewards is 0.05, I = 10 .e.g., the number of cluster heads and the number of rewards,are changed. • More Cluster Heads:
When there is only one rewardoffered to the cluster head that allocates the largestamount of CPU power, the cluster heads allocatemore CPU power when there are 5 cluster headsthan that of 15 cluster heads. For example, in Fig. 4,the cluster head with a valuation of . allocates . W when there are 5 workers but only allocates . W for computation when there are 15 workers.When there are more cluster heads participating inthe auction, the competition among the cluster headsis stiffer and the probability of winning the rewarddecreases. As a result, the cluster heads allocate asmaller amount of CPU power.However, similar trends are only observed for clusterheads with low valuations, e.g., v i = 0 . , whenmultiple rewards are offered. When the number ofcluster heads increases, the cluster heads with lowvaluations reduce their allocation of CPU power forthe CDC subtasks. However, this is not observed forcluster heads with high valuations, e.g., v i = 0 . .Figure 6, Fig. 7 and Fig. 8 show that the clusterheads with high valuations allocate more CPU powerwhen there are more cluster heads competing for themultiple rewards offered by the master. Specifically,when the master offers 4 rewards with a differenceof 0.05 between the consecutive rewards, the clusterhead with a valuation of . allocates W when there are 5 cluster heads but allocates
W whenthere are 15 cluster heads. When multiple rewardsare offered, since it is possible for the cluster headsto still win one of the remaining rewards even if theydo not win the largest amount of reward, i.e., topreward, the cluster heads are more willing to allocatetheir CPU power for the CDC subtasks. Hence, thecluster heads with high valuations allocate moreCPU power to increase their chance of winning thetop reward. • More Rewards:
When the number of cluster headsparticipating in the all-pay auction is fixed, the clus-ter heads allocate more CPU power when there aremore rewards that are offered. It is seen from Fig. 9that when there are 10 cluster heads in the all-payauction, the cluster head with a valuation of . allocates CPU power of W when 5 rewards areoffered as compared to
W and
W when 3and 4 rewards are offered respectively.
We compare the proposed two-level coalition-auction ap-proach against two other schemes, i.e., hedonic coalitionformation among workers with random allocation of CPUpower by the cluster heads and no coalition among workerswith random allocation of CPU power by the cluster heads.Figure 10 shows the comparison of the performance of theproposed two-level game-theoretic approach against otherschemes in a edge computing network with cluster heads. Round 1 Round 2 Round 3
Simulation Rounds T o t a l A m o un t o f C P U P o w e r A ll o c a t e d f o r C D C T a s k Two-stage Coalition-AuctionHedonic Coalition, Random CPU Power AllocationNo Coalition, Random CPU Power Allocation
Fig. 10: Comparison of the two-level coalition-auction ap-proach with other schemes.We observe that when coalitions are formed among workers,the total amount of CPU power allocated for the CDCsubtasks is generally higher. This is because by formingcoalitions with the workers, the cluster heads have moreavailable CPU power to perform their computation tasks.Instead of randomly allocating CPU power for the CDCsubtasks, the cluster heads are incentivized to allocate moreCPU power when the master offers homogeneous rewardsto the cluster heads under an all-pay auction. The averageof the total amount of CPU power allocated for the CDCtask under the two-level coalition-auction approach and thecoalition with random allocation scheme are
W and
W respectively.
ONCLUSION
In this paper, we proposed a two-level framework thatincentivizes cluster heads and workers to contribute CPUpower to facilitate the CDC tasks. The master applies thepolynomial codes to divide the dataset and allocate theCDC subtasks to the cluster heads. In the lower level, wepropose a hedonic coalition formation game in which eachworker chooses its coalition based on its individual utility.In the upper level, we design an all-pay auction to incen-tivize cluster heads, given their coalitions of workers, toparticipate in the CDC tasks by contributing larger amountof CPU power. Specifically, we use the recovery thresholdachieved by the polynomial codes to determine the numberof rewards to be offered in the all-pay auction. Then, themaster determines the reward structure to maximize itsutility given the strategies of the cluster heads. Simulationresults show that the utility of the cloud server is maximizedwhen it offers multiple homogeneous rewards to incentivizethe cluster heads to allocate more CDC power for the CDCsubtasks.In future work, we will consider workers with valuationschosen from probability distributions, other than uniform. R EFERENCES [1] J. Liu, H. Shen, H. S. Narman, W. Chung, and Z. Lin, “A Survey ofMobile Crowdsensing Techniques: A Critical Component for TheInternet of Things,”
ACM Transactions on Cyber-Physical Systems ,vol. 2, no. 3, 2018. [2] F. Kalim, J. P. Jeong, and M. U. Ilyas, “CRATER: A Crowd SensingApplication to Estimate Road Conditions,”
IEEE Access , vol. 4,pp. 8317–8326, 2016.[3] J. Dutta, C. Chowdhury, S. Roy, A. I. Middya, and F. Gazi,“Towards Smart City: Sensing Air Quality in City Based on Op-portunistic Crowd-Sensing,” in
Proceedings of the 18th InternationalConference on Distributed Computing and Networking , ICDCN ’17,(Hyderabad, India), Association for Computing Machinery, 2017.[4] S. Jovanovi´c, M. Jovanovi´c, T. Škori´c, S. Joki´c, B. Milovanovi´c,K. Katzis, and D. Baji´c, “A Mobile Crowd Sensing Applicationfor Hypertensive Patients,”
Sensors , vol. 19, no. 2, p. 400, 2019.[5] W. Shi and S. Dustdar, “The Promise of Edge Computing,”
Com-puter , vol. 49, no. 5, pp. 78–81, 2016.[6] J. Dean and L. A. Barroso, “The Tail at Scale,”
Communications ofthe ACM , vol. 56, no. 2, p. 74–80, 2013.[7] G. Ananthanarayanan, S. Kandula, A. G. Greenberg, I. Stoica,Y. Lu, B. Saha, and E. Harris, “Reining in the Outliers in Map-Reduce Clusters using Mantri.,” in , p. 24,2010.[8] J. S. Ng, W. Y. B. Lim, N. C. Luong, Z. Xiong, A. Asheralieva,D. Niyato, C. Leung, and C. Miao, “A Survey of Coded DistributedComputing,” arXiv preprint arXiv:2008.09048 , 2020.[9] M. F. Aktas, P. Peng, and E. Soljanin, “Straggler Mitigation byDelayed Relaunch of Tasks,”
ACM SIGMETRICS Performance Eval-uation Review , vol. 45, no. 3, pp. 224–231, 2018.[10] M. F. Aktas, P. Peng, and E. Soljanin, “Effective Straggler Mitiga-tion: Which Clones Should Attack and When?,”
ACM SIGMET-RICS Performance Evaluation Review , vol. 45, no. 2, p. 12–14, 2017.[11] M. A. Maddah-Ali and U. Niesen, “Fundamental Limits ofCaching,”
IEEE Transactions on Information Theory , vol. 60, no. 5,pp. 2856–2867, 2014.[12] N. Karamchandani, U. Niesen, M. A. Maddah-Ali, and S. N.Diggavi, “Hierarchical Coded Caching,”
IEEE Transactions on In-formation Theory , vol. 62, no. 6, pp. 3212–3229, 2016.[13] Z. Zhang, L. Cherkasova, and B. T. Loo, “Performance Modelingof MapReduce Jobs in Heterogeneous Cloud Environments,” in , (SantaClara, California, USA), pp. 839–846, 2013.[14] S. Li, M. A. Maddah-Ali, Q. Yu, and A. S. Avestimehr, “A Fun-damental Tradeoff Between Computation and Communication inDistributed Computing,”
IEEE Transactions on Information Theory ,vol. 64, no. 1, pp. 109–128, 2018.[15] F. Haddadpour, Y. Yang, V. Cadambe, and P. Grover, “Cross-Iteration Coded Computing,” in , (Mon-ticello, Illinois, USA), pp. 196–203, 2018.[16] K. Wan, M. Ji, and G. Caire, “Topological Coded DistributedComputing,” arXiv preprint arXiv:2004.04421 , 2020.[17] Q. Yu, M. Maddah-Ali, and S. Avestimehr, “Polynomial Codes: anOptimal Design for High-Dimensional Coded Matrix Multiplica-tion,” in
Advances in Neural Information Processing Systems 30 (NIPS2017) (I. Guyon, U. V. Luxburg, S. Bengio, H. Wallach, R. Fergus,S. Vishwanathan, and R. Garnett, eds.), pp. 4403–4413, CurranAssociates, Inc., 2017.[18] K. Lee, C. Suh, and K. Ramchandran, “High-dimensional CodedMatrix Multiplication,” in , (Aachen, Germany), pp. 2418–2422, 2017.[19] N. Raviv, R. Tandon, A. Dimakis, and I. Tamo, “Gradient Codingfrom Cyclic MDS Codes and Expander Graphs,” in
Proceedingsof the 35th International Conference on Machine Learning (J. Dy andA. Krause, eds.), vol. 80 of
Proceedings of Machine Learning Research ,(Stockholmsmässan, Stockholm Sweden), pp. 4305–4313, PMLR,2018.[20] S. Dutta, V. Cadambe, and P. Grover, “Coded Convolution forParallel and Distributed Computing within a Deadline,” in , (Aachen,Germany), pp. 2403–2407, 2017.[21] S. Wang, J. Liu, N. Shroff, and P. Yang, “Fundamental Limits ofCoded Linear Transform,” arXiv preprint arXiv:1804.09791 , 2018.[22] Q. Yu, M. A. Maddah-Ali, and A. S. Avestimehr, “Coded FourierTransform,” in , (Monticello, Illinois, USA),pp. 494–501, 2017.[23] S. Kiani, N. Ferdinand, and S. C. Draper, “Exploitation of Strag-glers in Coded Computation,” in sium on Information Theory (ISIT) , (Vail, Colorado, USA), pp. 1988–1992, 2018.[24] E. Ozfatura, D. Gündüz, and S. Ulukus, “Speeding Up DistributedGradient Descent by Utilizing Non-persistent Stragglers,” in , (Paris,France), pp. 2729–2733, 2019.[25] M. Dai, Z. Zheng, S. Zhang, H. Wang, and X. Lin, “SAZD: A LowComputational Load Coded Distributed Computing Frameworkfor IoT Systems,” IEEE Internet of Things Journal , vol. 7, no. 4,pp. 3640–3649, 2020.[26] Z. Ennya, M. Y. Hadi, and A. Abouaomar, “Computing TasksDistribution in Fog Computing: Coalition Game Model,” in , (Marrakesh, Morocco), pp. 1–4, 2018.[27] T. Zhang, “Data Offloading in Mobile Edge Computing: A Coali-tion and Pricing Based Approach,”
IEEE Access , vol. 6, pp. 2760–2767, 2018.[28] O. A. Wahab, J. Bentahar, H. Otrok, and A. Mourad, “TowardsTrustworthy Multi-Cloud Services Communities: A Trust-BasedHedonic Coalitional Game,”
IEEE Transactions on Services Comput-ing , vol. 11, no. 1, pp. 184–201, 2018.[29] C. Xu, K. Zhu, R. Wang, and Y. Xu, “Dynamic Selection of Min-ing Pool with Different Reward Sharing Strategy in BlockchainNetworks,” in
ICC 2020 - 2020 IEEE International Conference onCommunications (ICC) , (Dublin, Ireland), pp. 1–6, 2020.[30] N. Archak and A. Sundararajan, “Optimal Design of Crowdsourc-ing Contests,”
International Conference on Information Systems (ICIS)2009 Proceedings , p. 200, 2009.[31] K. Yoon, “The Optimal Allocation of Prizes in Contests: An Auc-tion Approach,” tech. rep., 2012.[32] J. S. Ng, W. Y. B. Lim, S. Garg, Z. Xiong, D. Niyato, M. Guizani,and C. Leung, “Collaborative Coded Computation Offloading: AnAll-pay Auction Approach,” arXiv preprint arXiv:2012.04854 , 2020.[33] J. Xiao, “Asymmetric All-pay Contests with HeterogeneousPrizes,”
Journal of Economic Theory , vol. 163, pp. 178 – 221, 2016.[34] Tie Luo, S. S. Kanhere, S. K. Das, and Hwee-Pink Tan, “OptimalPrizes for All-Pay Contests in Heterogeneous Crowdsourcing,” in , (Philadelphia, Pennsylvania, USA), pp. 136–144, 2014.[35] C. Cohen and A. Sela, “Allocation of Prizes in Asymmetric All-pay Auctions,”
European Journal of Political Economy , vol. 24, no. 1,pp. 123 – 132, 2008.[36] Z. Wen and L. Lin, “Optimal Fee Structures of CrowdsourcingPlatforms,”
Decision Sciences , vol. 47, no. 5, pp. 820–850, 2016.[37] F. Didier, “Efficient Erasure Decoding of Reed-Solomon Codes,” arXiv preprint arXiv:0901.1886 , 2009.[38] T. Luo, S. K. Das, H. P. Tan, and L. Xia, “Incentive MechanismDesign for Crowdsourcing: An All-Pay Auction Approach,”
ACMTransactions on Intelligent Systems and Technology , vol. 7, no. 3, 2016.[39] W. Saad, Z. Han, M. Debbah, A. Hjorungnes, and T. Basar, “Coali-tional Game Theory for Communication Networks,”
IEEE SignalProcessing Magazine , vol. 26, no. 5, pp. 77–97, 2009.[40] A. Bogomolnaia and M. O. Jackson, “The Stability of HedonicCoalition Structures,”
Games and Economic Behavior , vol. 38, no. 2,pp. 201 – 230, 2002.[41] Y. Zhang, J. He, and S. Guo, “Energy-Efficient Dynamic TaskOffloading for Energy Harvesting Mobile Cloud Computing,” in , (Chongqing, China), pp. 1–4, 2018.[42] Y. Hao, M. Chen, L. Hu, M. S. Hossain, and A. Ghoneim, “EnergyEfficient Task Caching and Offloading for Mobile Edge Comput-ing,”
IEEE Access , vol. 6, pp. 11365–11373, 2018.
Jer Shyuang Ng graduated with Double (Hon-ours) Degree in Electrical Engineering (HighestDistinction) and Economics from National Uni-versity of Singapore (NUS) in 2019. She is cur-rently an Alibaba PhD candidate with the AlibabaGroup and Alibaba-NTU Joint Research Insti-tute, Nanyang Technological University (NTU),Singapore. Her research interests include incen-tive mechanisms and edge computing.
Wei Yang Bryan Lim graduated with doubleFirst Class Honours in Economics and BusinessAdministration (Finance) from the National Uni-versity of Singapore (NUS) in 2018. He is cur-rently an Alibaba PhD candidate with the AlibabaGroup and Alibaba-NTU Joint Research Insti-tute, Nanyang Technological University, Singa-pore. His research interests include FederatedLearning and Edge Intelligence.
Zehui Xiong (S’17) received his B.Eng degreewith the highest honors in TelecommunicationEngineering from Huazhong University of Sci-ence and Technology, Wuhan, China, in Jul2016. From Aug 2016 to Oct 2019, he pursuedthe Ph.D. degree in the School of ComputerScience and Engineering, Nanyang Technolog-ical University, Singapore. Since Nov 2019, hehas been with Alibaba-NTU Singapore Joint Re-search Institute. He was a visiting scholar withDepartment of Electrical Engineering at Prince-ton University from Jul to Aug 2019. He was also a visiting scholar withBBCR lab in Department of Electrical and Computer Engineering at Uni-versity of Waterloo from Dec 2019 to Jan 2020. His research interestsinclude resource allocation in wireless communications, network gamesand economics, blockchain, and edge intelligence.
Dusit Niyato (M’09-SM’15-F’17) is currentlya professor in the School of Computer Sci-ence and Engineering and, by courtesy, Schoolof Physical & Mathematical Sciences, at theNanyang Technological University, Singapore.He received B.E. from King Mongkuk’s Instituteof Technology Ladkrabang (KMITL), Thailand in1999 and Ph.D. in Electrical and Computer Engi-neering from the University of Manitoba, Canadain 2008. He has published more than 380 tech-nical papers in the area of wireless and mobilenetworking, and is an inventor of four US and German patents. He hasauthored four books including “Game Theory in Wireless and Commu-nication Networks: Theory, Models, and Applications” with CambridgeUniversity Press. He won the Best Young Researcher Award of IEEECommunications Society (ComSoc) Asia Pacific (AP) and The 2011IEEE Communications Society Fred W. Ellersick Prize Paper Award.Currently, he is serving as a senior editor of IEEE Wireless Com-munications Letter, an area editor of IEEE Transactions on WirelessCommunications (Radio Management and Multiple Access), an areaeditor of IEEE Communications Surveys and Tutorials (Network andService Management and Green Communication), an editor of IEEETransactions on Communications, an associate editor of IEEE Transac-tions on Mobile Computing, IEEE Transactions on Vehicular Technology,and IEEE Transactions on Cognitive Communications and Networking.He was a guest editor of IEEE Journal on Selected Areas on Communi-cations. He was a Distinguished Lecturer of the IEEE CommunicationsSociety for 2016-2017. He was named the 2017, 2018, 2019 highly citedresearcher in computer science. He is a Fellow of IEEE.
Cyril Leung received the B.Sc. (First ClassHons.) degree from Imperial College, Univer-sity of London, U.K., and the M.S. and Ph.D.degrees in electrical engineering from StanfordUniversity. He has been an Assistant Professorwith the Department of Electrical Engineeringand Computer Science, Massachusetts Instituteof Technology, and the Department of SystemsEngineering and Computing Science, CarletonUniversity. Since 1980, he has been with the De-partment of Electrical and Computer Engineer-ing, University of British Columbia (UBC), Vancouver, Canada, wherehe is a Professor and currently holds the PMC-Sierra Professorship inNetworking and Communications. He served as an Associate Dean ofResearch and Graduate Studies with the Faculty of Applied Science,UBC, from 2008 to 2011. His research interests include wireless com-munication systems, data security and technologies to support agelessaging for the elderly. He is a member of the Association of ProfessionalEngineers and Geoscientists of British Columbia, Canada. Dong In Kim (S’89–M’91–SM’02–F’19) receivedthe Ph.D. degree in electrical engineering fromthe University of Southern California at Los An-geles, Los Angeles, CA, USA, in 1990. He wasa tenured Professor with the School of Engi-neering Science, Simon Fraser University, Burn-aby, BC, Canada. Since 2007, he has been withSungkyunkwan University, Suwon, South Korea,where he is currently a Professor with the Col-lege of Information and Communication Engi-neering. He has been elevated to the grade ofFellow of the IEEE for his contributions to the cross-layer design ofwireless communications systems. He is also a Fellow of the KoreanAcademy of Science and Technology and the National Academy of Engi-neering of Korea. He is a first recipient of the NRF of Korea EngineeringResearch Center in Wireless Communications for RF Energy Harvest-ing, from 2014 to 2021. From 2001 to 2014, he served as an Editor ofSpread Spectrum Transmission and Access for the IEEE Transactionson Communications. From 2002 to 2011, he also served as an Editorand a Founding Area Editor of Cross-Layer Design and Optimizationfor the IEEE Transactions on Wireless Communications. From 2008 to2011, he served as the Co-Editor-in-Chief for the IEEE/KICS Journal ofCommunications and Networks. He served as the Founding Editor-in-Chief for the IEEE Wireless Communications Letters, from 2012 to 2015.Since 2015, he has been serving as an Editor-at-Large of WirelessCommunication I for the IEEE Transactions on Communications.
Junshan Zhang (S’98-M’00-SM’06-F’12) re-ceived the Ph.D. degree from the School of ECE,Purdue University, in 2000. He joined the Schoolof ECEE, Arizona State University, AZ, USA,in 2000, where he has been a Professor since2010. His research interests fall in the generalfield of information networks and its intersec-tions with power networks and social networks,and fundamental problems in information net-works and energy networks, including model-ing and optimization for smart grid, optimiza-tion/control of mobile social networks and cognitive radio networks, andprivacy/security in information networks. Dr. Zhang is a recipient of theONR Young Investigator Award in 2005 and the NSF CAREER awardin 2003. He received the Outstanding Research Award from the IEEEPhoenix Section in 2003. He co-authored two papers that received theBest Paper Runner-Up Award of the IEEE INFOCOM 2009 and theIEEE INFOCOM 2014, and a paper that received the IEEE ICC 2008Best Paper Award. He was the TPC Co-Chair of a number of majorconferences in communication networks, including INFOCOM 2012,WICON 2008, and IPCCC’06, and the TPC Vice-Chair of ICCCN’06. Hewas the General Chair of the IEEE Communication Theory Workshop2007. He was an Associate Editor of the IEEE Transactions on WirelessCommunications, an Editor of the Computer Network Journal, and anEditor the IEEE Wireless Communication Magazine. He was a Distin-guished Lecturer of the IEEE Communications Society. He is currentlyserving as the Editor-in-Chief of the IEEE Transactions on WirelessCommunications, an Editor-at-Large of the IEEE/ACM Transactions onNetworking and an Editor of the IEEE Network Magazine.